In the last chapters, we led up to the idea of building a quantum computer using entangled particles as the underlying building blocks. If the goal is to run Grover’s search algorithm, then a quantum search engine probably already exists and can be purchased from D-Wave for $10 million. If, on the other hand, the goal is to have a truly universal quantum computer capable of running Shor’s factoring algorithm, in order to crack a 1024-bit public key, a quantum computer with all the bells and whistles and running full error correction would be a tera-qubyte machine, that is, 1012 qubytes or approximately 1013 qubits. e current record in an ion-trap quantum computer is around 13 qubits not 1013 qubits-we are 12 orders of magnitude away from a machine that would be of use for factoring such large numbers. Nevertheless, work continues apace in the study and development of ever-larger machines. A recent Intelligence Advanced Research Projects Activity program, in which I participate, proposes to estimate the resources, both quantum and classical, that a future quantum computer would need to solve a pantheon of dierent algorithms, not just factoring. To quote the head of the program, American scientist Mark Heiligman, of the Oce of the Director of National Intelligence, “National security decisions will now be made on the basis that there is no objective physical reality.”2 But as Nobel Laureate William Phillips is fond of saying, “e chances of building a quantum computer are 50-50: and by that I mean a 50% chance in 50 years.” What are we then to do in the meantime?
In the last chapters, we led up to the idea of building a quantum computer using entangled particles as the underlying building blocks. If the goal is to run Grover’s search algorithm, then a quantum search engine probably already exists and can be purchased from D-Wave for $10 million. If, on the other hand, the goal is to have a truly universal quantum computer capable of running Shor’s factoring algorithm, in order to crack a 1024-bit public key, a quantum computer with all the bells and whistles and running full error correction would be a tera-qubyte machine, that is, 1012 qubytes or approximately 1013 qubits. e current record in an ion-trap quantum computer is around 13 qubits not 1013 qubits-we are 12 orders of magnitude away from a machine that would be of use for factoring such large numbers. Nevertheless, work continues apace in the study and development of ever-larger machines. A recent Intelligence Advanced Research Projects Activity program, in which I participate, proposes to estimate the resources, both quantum and classical, that a future quantum computer would need to solve a pantheon of dierent algorithms, not just factoring. To quote the head of the program, American scientist Mark Heiligman, of the Oce of the Director of National Intelligence, “National security decisions will now be made on the basis that there is no objective physical reality.”2 But as Nobel Laureate William Phillips is fond of saying, “e chances of building a quantum computer are 50-50: and by that I mean a 50% chance in 50 years.” What are we then to do in the meantime?
Quantum computation is not an imminent achievement . We will not wake up tomorrow and find quantum computers on our desks. At the time of this writing, IBM has just unveiled the Condor – a chip containing 1,121 superconducting qubits. This number will likely increase (perhaps spectacularly) in the near future. However, the qubits have a relatively large error rate and, in the absence of a breakthrough advance that significantly lowers this rate, one would need hundreds of physical qubits to realize an error-corrected logical qubit. Predicting how long it will take to achieve the controlled manipulation of thousands of (logical) qubits (i.e., a number of qubits consistent with the “useful” implementation of known quantum algorithms) is probably not a rational undertaking, at least for a scientist. The most pessimistic experts regard the lively research activity in quantum computation as a waste of time and resources. But the pervading mood in this field is one of cautious optimism, particularly considering the possibility of achieving useful tasks (e.g., simulating quantum systems) with a sufficiently large number of “imperfect” qubits. Expediency may be responsible for some of this optimism; a good feeling about the prospects for quantum computation certainly helps when it comes to publishing papers and getting funded. Ultimately, however, the responsibility for the wise distribution of resources rests with the funding agencies; scientists, on the other hand, are called to explore every aspect of real and possible worlds that looks fascinating and worthy of their curiosity. From this perspective, quantum computation (defined broadly) offers plenty of opportunities for producing “good science.” The problems brought to the fore by the development of this field range from basic questions regarding the foundations of quantum mechanics and unsolved problems in computational complexity, to the investigation of new phases of matter (e.g., topological quantum phases), the quest for “complete” control over quantum systems, and the critical need for significant advances in materials science and device fabrication (e.g., engineering nanoscale heterostructures with atomic precision). Progress in these areas will naturally open new possibilities for interesting applications. Precisely identifying these applications is premature, but building a (general purpose) quantum computer will not necessarily be among the first of them.
The introduction of quantum computing has led to an immense increase in performance of various search and encryption algorithms. However the practical realization of quantum computers is1a challenging task. Hence quantum algorithms are being modeled on classical platforms. Hardware emulation of quantum computers has been found to be more effective than software simulations. In this paper, Grover's search algorithm has been modeled in the classical platform using both hardware emulation and software simulation approaches. The modified architecture of the field programmable gate arrays (FPGAs) and the unique data path implemented in this paper results in considerable speed-up and lesser utilization of resources as compared to other existing works. This work includes both simulation and emulation of Grover’s search for 2,3 and 5 qubit systems. The novel emulation approach has led to a 104times reduction in time of search than the corresponding simulation.
The goals of this paper are to show the following. First, Grover's algorithm can be viewed as a digital approximation to the analog quantum algorithm proposed in "An Analog Analogue of a Digital Quantum Computation", by E. Farhi and S. Gutmann, Phys.Rev. A 57, 2403 - 2406 (1998), quant-ph/9612026. We will call the above analog algorithm the Grover-Farhi-Gutmann or GFG algorithm. Second, the propagator of the GFG algorithm can be written as a sum-over-paths formula and given a sum-over-path interpretation, i.e., a Feynman path sum/integral. We will use nonstandard analysis to do this. Third, in the semi-classical limit ℏ→0, both the Grover and the GFG algorithms (viewed in the setting of the approximation in this paper) must run instantaneously. Finally, we will end the paper with an open question. In "Semiclassical Shor's Algorithm", by P. Giorda, et al, Phys. Rev.A 70, 032303 (2004), quant-ph/0303037, the authors proposed building semi-classical quantum computers to run Shor's algorithm because the success probability of Shor's algorithm does not change much in the semi-classical limit. We ask the open questions: In the semi-classical limit, does Shor's algorithm have to run instantaneously?
The dream of quantum computers has moved a small step closer to reality with the first successful demonstrations of quantum algorithms in experiments and a novel proposal to build a quantum computer based on the silicon technology used in existing computers. Quantum computers would exploit the ability of quantum particles to be in a superposition of two states at the same time and, for certain tasks, would have the potential to outperform classical computers by many orders of magnitude (Physics World March pp47–52).
It was Richard Feynman who first proposed, in 1982, the far-reaching concept of \na ”quantum computer”—a device more powerful than classical computers. The idea \nof a quantum computer is to employ the fascinating and often counterintuitive \nlaws of quantum mechanics to process information. It is far from obvious that \nthe proposed concept of a quantum computer is more powerful than its classical \ncounterpart, it was only in 1994 when Peter Shor theoretically demonstrated the \nexistence of a quantum algorithm for factorizing integers into prime factors \nthat runs in polynomial time unlike its classical counterpart which works in \nsub-exponential time. The factorization of integers into prime factors is the \nbasis of asymmetric cryptography. These early theoretical results lunched an \nimmense interest of the scientific community. Already during ’90s, the first \nproposals for the physical implementation of quantum computation emerged. Ever \nsince, many experimental groups around the world pursued different physical \nimplementations of quantum bits (qubits). The first decade of the new century \nsaw a steady improvement in the control and decoherence time (the time over \nwhich the information carried by the qubit is lost) for various qubits by many \norders of magnitude. The natural next step in this context is to answer the \nquestion of how to scale the system up to include many qubits and thus build a \nquantum computer? One of the main parts of this thesis addresses exactly this \nquestion, namely the question of architecture and scalability of future quantum \ncomputer. \n \nAmong various different physical realizations of qubits, the idea of using \nelectron spins trapped in electrostatic semiconductor quantum dots as the \nbuilding blocks of a quantum computer (the so-called spin qubits), put forward \nby Daniel Loss and David DiVincenzo in 1997, triggered tremendous interest in \nscientific community. Nevertheless, the implementation of the original \nLoss-DiVincenzo proposal posed a considerable technical challenge. It used \nquantum tunneling between qubits to enable their communication with each other, \nand thus required that the qubits to be placed very close to each other. This \nrequirement not only leaves little \nspace for the placement of the vast amount of gates and wirings needed to define \nthe electrostatic quantum dots, but also makes it challenging to control the \nlocal magnetic field needed for single-qubit operations. For these reasons, no \nsystem with more than a couple of spin qubits has been \nsuccessfully implemented thus far. In the first part of this thesis, we leap \nover this long-standing problem with an entirely different strategy of using \nmetallic floating gates or ferromagnets to couple together qubits that are \nseparated over a long distance. Our scheme works for any type of \nspin qubits, including the qubits based on nitrogen-vacancy center (NV-center) \nin diamond and technologically very important silicon qubits. \n \nThe main topic of this thesis is related to quantum computer. Still, quite \nunexpectedly, some of the ideas we employed in order to tackle the problem of \nquantum computer scalability can be utilized in a completely different field of \nresearch, namely, in the field of magnetic field sensing. Qubit are not only a \nnecessary ingredient of quantum computer but they also provide a way to measure \nvery accurately magnetic fields. The magnetometer build upon the qubit based on \nNV-center, so-called NV-magnetometer, emerged in recent years as most sensitive \nmagnetic moment sensor. These magnetometers are able to detect about hundred \nnuclear spins within a minute of acquisition time. In the second part of this \nthesis, we propose an entirely novel experimental realization of \nNV-magnetometers which increases present NV-center sensitivities by four orders \nof magnitude at room temperature. This unprecedented amplification of \nsensitivity will render magnetometers capable of detecting individual nuclear \nspins. This amplification is achieved by introducing a ferromagnetic particle \nbetween the nuclear spin that needs to be detected and the NV-magnetometer. Our \nsetup, in contrast to existing schemes, is particularly advantageous because, \ndue to the large amplification of sensitivity, the nuclear spin need not lie \nwithin a few nanometers of the surface but rather can be detectable at a \ndistance of 30 nm. With these novelties, our scheme provides chemically \nsensitive spin detection under ambient conditions allowing nanoscale resolution \nof nuclear magnetic moments in biological systems—the holy grail of nuclear \nmagneticresonance. \n \nIn the last part of the thesis we focus our attention to a new direction in \nquantum computer implementation that deals with topological quantum computer \nintroduced by Alexei Kitaev in 1997; in this approach the idea is to use \nquasiparticles with ”fractional” statistics and to perform the single- and \ntwo-qubit gates by merely exchanging these quasiparticles. Additionally, \ninformation in this system is stored non-locally thus it mitigates the problem \nof decoherence caused by local noise from the environment. Majorana fermions are \none of the most well known examples of such excitations. We analyze transport \nsignatures of different topological states in one-dimensional systems, like \nMajorana fermions and fractionally charged states. We envision an Aharonov-Bohm \nsetup wherein conductance measurement provides a clear signature of presence of \nfractionally charged fermionic states, since oscillations with double period \nemerge in this case. Additionally, we propose a very simple setup that enables \nexistence of degenerate mid-gap states, so-called Tamm-Shockley states that are \ncharacterized by fractional charge and discuss possible ways of detecting these \nstates experimentally. \n
Abstract Quantum superposition is a powerful computational tool to have at one’s disposal – in fact, a phrase along the lines quantum computers evaluate a function for every possible input in superposition and hence achieve a massive speed-up appears in virtually every popular and media article on the subject. The reality is, however, a little more complicated. In this chapter, we look at what is probably the closest quantum algorithm to the naive idea of ‘evaluating for every possibility at once’, namely Lov Grover’s quantum search algorithm. Grover’s algorithm is one of the oldest and most important quantum algorithms and offers a quadratic advantage over the best possible classical algorithm for the problem of searching an unstructured database. Notably, a quadratic quantum advantage is the best one can hope for in certain problems that exhibit a lack of structure, which is proven by considering the performance of the best possible quantum algorithm tackling the OR problem.